The kinetic energy of a system of N particles is the sum of the kinetic energy of the particles of the system:

When the particle system is a rigid body that is rotating with respect to an axis that passes through its center of mass, the particles of the solid describe a circular motion with respect to that point:

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In this situation we can decompose the kinetic energy of the solid into two terms: one that describes the kinetic energy of translation of its center of mass and another that describes the kinetic energy of translation of the particles of the solid with respect to the center of mass:

And since the movement of the particles of the solid with respect to the center of mass is circular, the relation between its linear velocity and its angular velocity is:

Substituting in the second summation of the kinetic energy we obtain the kinetic energy of rotation:

As we will see in the problems, when a solid is rotating we must take this term into account when writing its kinetic energy. If in addition the center of mass of the solid is in movement, we must include its kinetic energy of translation.

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